In the Standard Model, fermion number is not conserved. Lepton number is conserved, because of an accidental symmetry. One cannot write down a renormalizable, gauge and Lorentz invariant operator that violates lepton number conservation in the Standard Model.
A Majorana neutrino would violate lepton number conservation by two units. To see this, consider, e.g. neutrinoless double beta decay. You can draw Feynman diagrams with two W−e−ve vertices in which two incoming W-bosons each decay into an electron and an electron-neutrino. The two electron-neutrinos annihilate (possible because they are Majorana particles), leaving a final state of two elecrons, violating lepton number conservation by 2 units.
You can see that Majorana neutrinos violate lepton number conservation by 2 units from the mass term. The mass term,
L=12mψTC−1ψ,
is not invariant under the
U(1) lepton number symmetry,
ψ→exp(iLθ)ψ. It picks up a phase of twice the lepton number of the neutrino, i.e.
ΔL=2 rather than
ΔL=0. A Majorana neutrino cannot be charged under a
U(1) symmetry.
Because there is not a lepton number U(1) symmetry, there is no conserved Noether charge corresponding to lepton number.
This post imported from StackExchange Physics at 2014-04-13 14:48 (UCT), posted by SE-user innisfree